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LIBRARY 

01     THK 

UNIVERSITY  OF  CALIFORNIA. 

(  )K 


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Received        G2,c3f~  . 

Accession  No.  7^-3  bO     .    Clots  No. 


ARITHMETIC 


SHORT  METHODS 


*  SWEET  * 


OF  THE 

UNIVERSITY 


SWEET'S 

Hand     Book 


OF 


SHORT  METHODS 


Arithmetic 


j.   s.   SWEET,   A.  M., 

'Principal  of  the  Santa  Rosa  IJusiness  College.    Santa    Ho.-a,  Cal. 

formerly    President  of   the    Oix-^oii  State  Xormal  School. 

Ashland.  Or.,  Author  of  Sxveet '-  Sv~u-m    of   Act- 

nal  Hnsiness  Practice,  Element*  of  Geom- 

etrv,  lousiness  t"orm~.  Etc. 


SANTA  ROSA,  CALIFORNIA 


Entered  according  to  Act  of  Congress,  in  the  yi-;ir  IS!)!!. 

ByJ.  S.  SWEET, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington,  D.  C. 

>. 

J 


PREFACE. 


The  principal  object  of  this  little  work  is  to  place  in 
the  hands  of  the  student,  in  compact  form,  many  of  the 
briefer  methods  of  rapid  calculations.  ''Time  is  money," 
and  especially  so  to  many  of  our  young  people  who  are 
trying  to  obtain  a  business  education  in  a  brief  time  and 
with  limited  means. 

Hoping  that  many  may  profit  by  the  suggestions  here- 
in contained,  I  most  respectfully  dedicate  this  little  volume 
to  the  young  business  people  of  America. 

Santa  Rosa,  Calif.,  1893.  J.  S.  SWEET. 


6  XIIORT  METHOD* 

2.     Slims  Greater  than  9. 

54321543 

56789678 


6543       654 

6789       789 


765     76     87 

789     89     89 


8  9 

9  9 


.V.     To  Itetrtl  fit  Sight. 

When  a  student  sees  the  figures  1  and  3  written  side  by 
side,  he  instantly  recognizes  "thirteen"  or  "thirty-one"  ac- 
cording to  their  positions.  The  same  facility  may  be  ac- 
quired in  regard  to  numbers  in  addition;  thus,  4  over  or 
under  8,  may  be  read  "twelve"  as  readily  as  the  figures  1 
and  2  side  by  side.  Ten  minutes  practice  daily  for  one 
month  will  accomplish  the  work. 

4.  Always  add  TWO  or  MORE  figures  at  a  time.  Never 
be  guilty  of  adding  single  figures.  Name  the  results  of  the 
following  as  rapidly  as  possible  : 

246975634674-89 
35323678988723 

38765725475399 

48797999888789 


IX  AltlTHMKTIC. 

56854322462537 

49987873878689 
73776298897778 
84698549739894 


»J.     Nine  added  to  any  number  is  always  ONE  LESS  in  its 
unit's  place  than  the  number.      Thus, 

8  —  •  9  -    7    in  its  unit's  place. 
36        9        5 

6*.     Eight  added  to  any  number  is  TWO  LESS  in  its  unit's 
place  than  the  number.     Thus, 

7        8  ==  15,         15-8  =  23. 
T.     To  Add  btf  TenM. 

A  good  method  is  to  add    by  10's,  carrying    the    EXCESS 
in  the  mind,  as  in  the  following : 

8"  72 

9  5 
63  95 
7  6 

30  27 

Here  the  3  of  the  13  is  carried  to  the  7  of  the  17  mak- 
ing three  tens  in  all.     Add  in  this  manner  the  following: 


3 

9 

6 

5 

9 

8 

8 

8 

8 

8 

7 

5 

5 

7 

9 

9 

9 

5 

9 

6 

4 

3 

4 

4 

5 

6 

4 

9 

8 

6 

8  *1IORT  MKTHOIt* 

8.     When  the  Columns  are  Long. 

When  there  are  two  or  more  columns  of  consider- 
able length,  add  each  column  separately  as  instructed,  and 
write  the  sum  of  each  alone,  then  combine  results  into 
one  number,  as  follows  : 

32476 
58976 
76892 
39428 
73548 
67943 
28745 
"^8 
37 
46 
43 
33 


378008 

This  method  is  almost  indispensable  in  book-keeping,  as 
an  error  can  be  located  much  more  readily  than  when  the 
separate  results  are  not  known. 

9.    To  Add  Two  Columns  at  a  Time. 

To  add  two  columns  at  a  time  practice  on  the  fol- 
lowing, by  adding  the  tens'  column  first,  and  by  reading  the 
units'  column,  tell  at  a  glance  the  number  to  carry  : 


23 
36 

72 
49 

35 
44 

66 
27 

38 
79 

38 
44 

59 
71 

88 
64 

39 

89 

88 
26 

86 
49 

94 

87 

75 

89 

85 
94 

f.\  ARITHMETIC.  9 

10.     Proof*  of  Addition. 

In  long  columns  the  best  proof  is  to  add  them  again,  up 
or  down,  the  opposite  of  your  first  addition.  In  short  col- 
umns and  several  of  them  to  add,  you  may  prove  the  work 
by  casting  out  the  9's  as  shown  below. 

25189654  -  4 
36972105  -  6 
94375517  -  5 
15155815  -  4 
85310652  -  3 
95315175  -  0 
352318918  -  4 

Casting  out  the  9's  of  the  first  nnmber,  we  have  an  e.< 
of  4 ;  of  the  second,  6 ;  of  the  third,  5  ;  and  so  on,  finally 
casting  out  the  9's  of  these  results  which    gives  an  excess  of 
4.     Also  by  casting  out  the  9's  of   the  sum,  we  have  4,  we 
therefore  conclude  that  the  work  is  correct. 

XOTK.  This  is  not  always  a  sure  test,  the  answer  mi_<rht  he  wrong1  and 
vet  prove  bv  this  method. 


SUBTRACTION 


11.  When  the  forty-five  combinations  treated  of  in  Ad- 
dition are  thoroughly  memorized,  the  process  of  subtraction 
is  a  very  simple  one.  This  consists  of  being  able  to  discern 
at  a  glance  the  digit  which  will  combine  with  one  of  those 
given  to  produce  the  other.  Thus, 

8 
3 

are  given,  and  the  question  is :  what  number  combines  with 
3  to  produce  8?  The  process  is  nearly  the  same  as  in  ad- 
ding, the  only  difference  is  that  we  must  furnish  one  of  the 
numbers  to  the  combination,  the  result  already  being  known. 
Read  the  differences  as  rapidly  as  possible  : 

98767896757988 
43234454326325 

15         16         17         14         13         12         18 

8986789 

Daily  drills  in  both  addition  and  subtraction  should  not 
be  neglected.  The  process  of  this  method  is  very  simple 
and  is  readily  learned.  Practice,  only,  will  perfect  it  and 
give  value  to  it. 


MULTIPLICATION 


7V.  With  Multiplication  we  begin  our  Slinrt  Methwls. 
supposing  the  student  to  be  sufficiently  advanced  to  know 
the  multiplication  table  to  the  12's.  If  not,  he  should  learn 
the  following 

MULTIPLICATION   TABLE: 


1 

•> 

3 

4 

5 

6 

7 

o 

§ 

10 

11 

12 

•2 

4 

o 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

OD 

60 

6 

12 

18 

24 

30 

42 

:>4 

60 

66 

72 

, 

14 

21 

28 

42 

4U 

56 

!63 

70 

77 

84 

- 

16 

Mi' 

40 

48 

56 

64 

80 

88 

96 

9 

18 

27 

45 

-")1 

68 

72 

81 

90 

99 

108 

10 

20 

40 

:,d 

C)() 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

.  i 

88 

99 

110 

121 

132 

1-2 

•24 

36 

48 

60 

12 

84 

96 

108 

120 

132 

144 

//>.     The  following  squares  of   numbers  should  also  be 
memorized  : 


12  XI10RT  METHODS 

13  X  13  ==  169  19  X  19        361 

14  X  14  =  196  20  X  20  ==  400 

15  X  15  =  225  21  X  21  =-  441 

16  X  16  —  256  22  X  22  ==  484 

17  X  17  ==  289  23  X  23  —  529 

18  X  18  =  324  24  X  24        576 

25  X  25  =    625 

14.  To  multiply  any  number  const stint/ 
of  two  digits  by  11. 

RULE.     \Vrite  the    *inn    of    the    digit*    between 
ilieiti,  ilie  number  fltii*  e.vpre^ned   ?'s  I  lir  product. 

EXAMPLES. —  11  times  24  =  264, 
11  "  36  ==  396, 
11  "  57  ==  627. 

XOTK.      \Ylu-ii  tin-it- -tun  is  10  or  inure,  carry  one  to  tin-  liutulivd's  di»-it. 

EXERCISES. 

15.  1.   Multiply  45  by  11.     4.   Multiply  75  by  11. 

2.  38  by  11.     5.  96  by  11. 

3.  92  by  11.     6.  88  by  11. 

16.  To  multiply  any  number  by  11. 

RULE.     Write  the  unit's   iigurc;    next, -write  the 

sum  of  the  ^l-nits  and  tens,  tJifit  i  lie  ^nin  of  1/ir  fi-n* 
and  hundreds,  etc.,  writing  flic  left  IKI'IK/  figure 
last,  carrying  i^lien  uec:es?:«r\'. 

EXAMPLE.—  11  times  12345  ==  135795. 

5 

4    j    5  9 

:i        4  -  7 

2        3  5 

1-2          3 
1 


7.Y  ARITHMETIC.  13 

EXERCISES. 

/;.  7.  Multiply  663  by  11.  4.  6731  by  11. 
938  by  11.  .7.  9884  by  11. 
734  by  11.  6.  72596  by  11. 

18.     To  multiply  by  22,  33,  etc. 

Rl'LK.  Mil  hi  ply  h\>  11  //>-  <i}>,>r<>,  ami  rhru  !>\'  ?,  .V, 
<>r  4,  etc. 

EXAMPLE.—  22  times  234  =.  2574  X  2  =  5148. 

XOTK.     The  work  should  be  done  mentally,  only  results  being;  written. 

EXERCISES. 

W.  7.  Multiply  64  by  22-  4.  374  by  55. 
65  by  33.  .J.  874  by  66. 
46  by  44.  V.  336  by  77. 

20.  To  multiply  by  ant/  tuttnbet*  betirecn 
12  and  2O. 

RULE.  Multiply  by  the  unifs  figure  only,  i.'rir- 
mg  the  r<\<it/r  «uJt>riln>  innitlx>r  <m<l  <mr  ~p}<irt>  r<> 
tin'  n'»-hr,  thru  ,i,J<l. 

.KS. — 13  times  24        24 


312  An*. 
14  times  175        175 

700 
2450  An». 

EXERCISES. 

21.     1.    Multiply  262     by  13  .7.   9624     by  17 

382     by  14.  6.   32694  by  18, 

497    by  15.  7.   27314  by  19. 

.;.  1824  by  16.  8.   98794  bv  12. 


22.     To  multiple/  by  21,  31,  41,  fil,  etc. 


RULE.  Multiply  by  the  tens  only,  i^n'ti'tig  Hie 
result  render  ilie  ninuhe  r  <i  n  <1  our  pluee  1o  ilie  left, 
then  add. 

EXAMPLE.  —  31  times  24          24 

72 
744 

EXERCISES. 

23.  1.     Multiply  35     by  31.          k     728    by  51. 
&  46    by  41.         5.     3824  by  61. 
3.  245  by  21.          6.     8452  by  71. 

24.  To  multiplt/  by  lo. 

RULE.  Anne.\-  one  eipli.er  /<>  ////'  number  <nnl  <i<I<t 
its  half. 

EXAMPLES.  —  15  times  28  =  280 

V2  of  280  =    140 

420 

15  times  35  =  350 
175 
525 

EXERCISES. 

25.  1.   Multiply  44    by  15.         4.   248       by  15. 

87    by  15.         5.   7634    by  15. 
3.  394  by  15.          6.   98768  by  15. 

20.     To  multipJt/  by  XI. 

RULE.      Take  one-half  the  nmnber    <tnd    write    it 

two  places  to  the  left  <nul  («J<!. 


_ 

flTNlVERSITY  ) 

IX  ARITHMETIC.  15 


KXAMPLKS. —  51  times  72  =       72 
V2  of  72        36 


4372 

51  times  45  =  45 

225 
2295 


EXERCISES. 

1.  Multiply  78  by  51.  4.  1384  by  51. 
324  by  51.  ,5.  4633  by  51. 
723  by  51.  6.  78254  by  51. 

28.     To  sqttftre  ft  n  tun  her  tr/iose  tntft  fiy- 
ttre  is  ,7. 

RULE.      Multiply  rite  /-»•;/>-'   <ligjr     />v    «i 
<t  11  (1  ' 


EXAMPLE. —  25  times  25  =  625. 

2  times  3        6,  annex  25  625. 

EXERCISES. 

1.  Multiply  35     by  35.  -7.    75  by  75. 

'  45     by  45.  6.  85  by  85. 

55     by  55.  ?.   95  'by  95. 

4-  65    by  65.  8.   105  by  105. 

•V0.     To  find  the  prodttct  of  two  nttnthers 
units'  (I if/ its  nre  ,7's. 


RULE.     To  rJir    product    of  the  tens  add  one-half 

tln'ir  :«uin  mxl  <nnn\\-  25  if 


XOTK.     Fractions  of  one-half  are  dropped. 


X6  SHORT 

EXAMPLES. —  25  times  45  -1125. 

1/2  of  (2  +  4) -h  2_X  4  =  11,  annex  25  =  =  1125. 

25  times  35  =875. 
1/2  of  (  2  -h  3)  -f-  2  X  3  ==  8,  annex  75        875. 

NOTK.     -  plus  1?  is  odd. 

EXERCISES. 

/>'/.     J.   Multiply  25     by  65.         4.  45  by  35. 

2.  25    by  85.         5.   65  by  35. 

3.  105  by  25.         H.   75  by  65. 

32.  To  find  the  product  of  two  numbers 
whose  tens9  digits  are  identical  and  the  sum 
of  the  units9  digits  is  10. 

RULE.  Multiply  the  tens'  digit  by  OTIC  greater 
and  annexe  the  product  of  the,  unit*'  <l,igi1*. 

EXAMPLE. —  43  times  47  =  2021. 
4X5  and  annex  7X3  ==  2021. 

EXERCISES. 

33.  1.   Multiply  29  by  21.         5.   38  by  32. 
2.  28  by  22.          n.   37  by  33. 

27  by  23.          7.   49  by  41. 

4.  39  by  31.         8.   48  by  42. 

34.  To  find  the  product  of  two  numbers 
whose  tens9  digits  are  consecutive,,  anil  the 
sum  of  the  units9  digits  is  10. 

Rl'LK.  To  (lie  product  of  tin*  /<>ss  ten*  and  one 
more  than  the  grfutcr,  <nni<'.\-  tin-  complement  of 
the  square  of  the  greater  number' *  unit  figure. 

XOTK.      Complement  of  ;i  munher  i-   TOO  les^  the  numlu-r. 

EXAMPLE —  87  times  73        6351. 

7    <  9        63;  complement  of  the  square  of  7     =  51; 
annex  it  to  63         6351. 


IX  ARITHMETIC. 


EXERCISES. 


Multiply  47  by  33. 
56  by  44. 
64  by  56. 


-9  by  71. 
(>.   84  bv  76. 


#6*.     To  jitttt  the  product  of  tiro  numbers 
tcheti  their  fats'  (lif/its  are  the  sattte. 


Rl'LK.      Ta,he  the  prmluct   <>f  tin'   unir*,    ne.\~r    rhe 
print  ncr  ;/s  riling  the  sum  <>f  tli-e  unit*,  then 

flu'  pr»>J  i/  cr  nf  i/ir  rcn*,   always  r,i  ~rr\-i  m?   rli*'    r»'7;s, 
if  itii  v. 


EXAMPLE. — 


73  times  75 
5        3 

8"  •    7 
7        7 


5475 

15  write  5,  carry  1 
56    carry  5. 
4V) 
5475 


EXERCISES. 


/.  Multiply  74  by  72. 
85  by  83. 
67  bv  65. 


4.  97  by  94. 
.5.  88  by  89. 
'/.  79  bv  78. 


•V#.     To  /r  tt  ft  the  product  of  two  tttrtttbet's 
trhett  the  units9  digits  are  iflettfieftt. 


Rl'LE.      Talif    r/i.,'    pnnlitcr  of   the   nn.it  ^ 
the  sum  <>f  t  lir  ten*  times  the  liuit*,  mul  the  product 
of  the  tens,  currying  i^'In'n  necessary. 


KXAMPLK.  —  44  times  74        3256. 


EXERCISES. 


/  Multiply  46  by  56. 
54  by  34. 
43  bv  53. 


4-   73  by  63. 

6.   87  by  47. 
n.   98  bv  28. 


OF  THE 


i8  XIIORT    METHODS 

40.  To  find  the  product  of  fntt/  two  num- 
bers consisting  of  two  digit*. 

RULE.  Take  the  product  of  the  units,  th-e  sum  of 
the  products  of  each  ten  times  the  other  unit,  and 
the  product  of  the  tens,  carrying  if  necessary. 

EXAMPLE.  —  47  times  36. 

6  X  7  ~-  42 

6X4       a    <"  7  =        45 
4X3=  12 

1692 
EXERCISES. 

41.  1.   Multiply  35  by  27.         4.   68  by  34. 
2.  47  by  34.         5.   78  by  46. 

52  by  46.         6.   39  by  35. 

42.  To  find  th  e  produ  ct  of  numbers  wh  <>  1  t 
one  part  of  the  multiplier  is  a  factor  of  the 
other. 

RULE.  Multiply  by  the  factor,  then  tliit?  product 
by  the  quotient  of  the  factor  into  tin-  oilier  part, 
<i  tnl 


EXAMPLE.—  231 

183 

Multiply  by  3  =  693 

"      this  product  by  6  =  4158 
42273 

423 
126 

Multiply  by  6  =  2538 

"      this  product  by  2  =  5076 


53298 
EXERCISES. 

43.  L     Multiply  1247  by  255. 

2.  792     by  279. 

3635  by  1089. 


IX     ARITHMETIC,  19 


44.     Jty  tnnltijrtu  ft//  the  factors  of  ft  num- 
ber. 

RULE.     Multiply  by  one  factor  <iml  ///  /.-• 
other. 


EXAMPLE.  —  21  times  65  ==  7  times  65  =    455 
and  455        3        1365. 

EXERCISES. 

4X.      1.   Multiply  73     by  42.         4.  97  by  14. 

83     by  35.         5.  87  by  36. 

123  by  27.          6.  79  by  49. 

46.     ToiHHltijrti/bt/  1O,  WO,  10OO,  etc. 


Annex  as  tna>ny  cipln'r*  <i*    there  ure    in 
1  lie   in  it  It  i  pi  ier. 

EXAMPLES.  —  10  times  76       -  760. 

100  times  125        12500. 


4;.     To  nitdtiptt/  ft//  ft  nu  multiple  of  1O. 
10O,  WOO,  etc. 


RULJZ.      Multiply  by  the  digital  number  and  tlieji 
<t  n  n  e.\-  ciphers. 

EXAMPLE. —  400  times  123          49200. 
2000  times  243  =  486000. 

48.     To    inultinlt/    ft//   .9,   or  fftit/   number 
of  ,9'x. 

RTLK.     Annex   11$   many  ciphers  as  there  &re  9* s 

and  finhtrarr  thr   nuni-her  multiplied. 

EXAMPLES. —  9  times  435  =  4350  —  435  =  3915. 
99  X  267  =  26700  —  267  ==  26433. 


20  ^IIORT    MKTIIOD* 

EXERCISES. 

49.  I.   Multiply  47     by  9.          Jt.   148  by  9.      - 
2  125  by  9.         5.   725  by  99. 

238  by  9.  '       6'.   675  by  999. 

50.  To  multiply  btj  any    number  endint/ 
in  9. 

RULE.  Multiply  by  ////•  n<'.\-f  greater  number 
and  from  the  product  subtract  the  number  multi- 
plied. 

EXAMPLE.— 382  times  49  =  382  X  50  —  382. 

382 

50 

"19100 

382 

18718 

EXERCISES. 

,7/.     1.   Multiply  128  by  69.          3.   326  by  599. 
2.  245  by  59.          //.   262  by  499. 

32.  To  multiply  by  ant/  number  a  little 
less  or  a  little  greater  than  100,  1000,  etc. 

HULK.  JLnnex  as  ma-ny  ciphers  «^  liters  a  re  fi^- 
nrc*  in  th.c.  multiplier  <nn1  mihtruci  <>r  <nld,  the  pr<>- 
<hn;t  of  tlic  difference  between  10O,  10()(),<>t<\,  ami 
(lie  multiplier. 

EXAMPLE.—  423  times  996        423000  —  4   ><  423. 

._    1(51)2 
421308 

EXERCISES. 

,-T.V.     L   Multiply  993  by  624.  J.   9994     by  425. 

0.        997  by  529.  <i.  9998  by  827. 

992  by  895.  7.  99993  by  963. 

.',.        326  hv  104.  8  1003  by  724. 


": 


IX     ARITHMETIC.  21 

34.     To   multiply   by  any   multiple   of  1), 
t  e.rceed  i  ny  i)O. 


RULE.     ^Multiply    b\'     the    multiple    <>/    mi 

tlnni.  t  he  git  'ru    nt-u  fri pi  irr,  and  £iii>tr<'icr  ?'rs 
ne-tenth. 

EXAM  PI. K.—  454  times  72 

454 

80 

36320  product  by  80 

3632         "        "       8 


32688        "        "    72 


EXERCISES. 

,7,7.  /.  Multiply  4(5  by  18.  5.  288  by  54. 
75  by  27.  r>.  384  by  63. 
82  by  36.  7.  772  by  75. 

4.  144bv  45.         8.   1244  bv  81.5 

OI 

.76*.     To  multiply  by  complements. 

P'roiu  eitliLT  number  subtract  the  comf>li' 
of  the  Other,    <tinl    <nnn\\-    t/n>    pr<nlucr    <tf-  t 
com 


XOTK.     Tlie  product  should  have  ;i~  mail  v  figure-  as  are  in    both    nuiii- 
-S:  sup]>ly  oi]ihi-rs  To  maki-  tht-m  the  same. 


p]>ly  oi]i 

EXAMPLES.  —  94  comp.  6  999  comp.  1 

97  comp.  3  999  comp.  1 

9118  998001 

A'd 

EXERCISES. 

,7;.  /.  Multiply  92  by  87.  4.  996  by  995. 
94  by  75.  5.  993  byi£9\L. 
99  by  93.  n.  998  by  895. 


22  SHORT    METHODS 

58.     To  find  the  product  of  two  tt  a  in  hers, 
each  of  which  is  a  little  over  1OO. 

RULE.     From  the  sit-in,  of   the    numbers   snl>rr<ict 
100  (nid  (.inn-cjc  the  product  of  tltc  ejccesses. 


EXAMPLE.—  115  times  104  ==  11960 
115  -f  104  —  100  ==  119 
To  119  annex  15  X  4  ==  11960. 

EXERCISES. 

39.     1.   Multiply  114  by  105.          4.   144  by  107. 

2.  122  by  103.          .7.    160  by  106. 

135  by  102.          6.   138  by  108. 

XOTH.     Applv  The  same  principle  to  the  following: 

1.  Multiply  1008  by  1007.         8.   1250  by  1003. 

2.  1125  by  1004.         4.   1475  by  1002. 

60.  To  find  the  prod  act  of  two  n  ambers 
one  of  which  is  more  ami  the  other  Jess  than 
100. 

RULE.     From,  tlic  srrm   of   tlir    nnn^hcr^    *ubtr<ict 
100,  aimex  two  <~ip//rr*  <ni<l  ^ul>f-r<ict  1  lit-  product  of 
the  excess  and 


EXAMPLE.  —  108 
98 

8  excess. 
2  complement 

10600 
16 
10584 

EXERCISES. 

61.     1.   Multiply  102  by  94.         4.   125  by  92. 

&  103  by  97.         5.   112  by  99. 

115  by  96.          6.   116  by  95. 

XOTK.      Apply  The  same  principle  to  the  following; 

1.  Multiply  1004  by  92.         S.   1015  by  92. 

2.  1008  by  95.         4.   1025  by  96. 


IX  ARITHMETIC. 


ALIQUOT  PHRTS. 


TABLE. 


y2  of  100  ==  so 

Vs  "  -  33^3 

V4  "  =  25 

Vs  "  =  20 

Ve  "  -  16% 

VT  "  - 


Vs  of  100  =  12% 

1/9          "  =  111/9 

Vio     "         -  10 


Vl2 
VlG 


of  100  =  37  tf 
=  62*4 

-  87y2 

-  66^3 

-  83*-3 

6       "  ^   1834 


s/16of  100-  31 H 

7/16         •'  -4334 

9/i6       {>         -  56M 

11/16      "  =  6834 

13/16      "  81J4 

9334 


1>2.     To  multiply  by   an  rtfiquot  part  of 
100. 


Anin\\-  ti.'o  ciphers,  </ 

inn!  multiply  fry  the  ntnncrafor  of  t  hr  frac- 
tional f<n~t  It  /.-  of  100. 

EXAMPLE-.  —  50  times  12  =  7200  -r-  2  =  3600. 
162  3  times  84        8400  -*-  6  ==  1400. 


EXERCISES. 

1.  Multiply  48  by  25 
33%  by  24. 
35  by  20. 

4.  63      by  14-1- 


5.  184    by  12i  o. 

6.  960     by  8V3. 

7.  3603  by 

8.  2560  by 


24  SHORT 

1.  Multiply  72  by  37V2.          4-  423  by  66%. 

2.  56  by  12V2.         -5.   144  by  83V3. 
96  by  87V2.          tf.   216  by  18%. 

fl-/.  To  multiply  by  10  times  an  aliquot 
part  of  100. 

RULE.  Annex  three  ciphers  <m<l  proered  ^s  In-- 
fore. 

Ex.—  166%  times  84  =  84000  4-  6  ===  14000. 

times  144  =  144000   -  12  -     12000. 

EXERCISES. 

6V>.     J    Multiply  125     by  48.         5.   112  by  62V2. 
0.  1236  by  3331/3.  4-   192  by  83V3. 

66.  To  multiply  by  (t  little  more  or  a  lit- 
tle Jess  than-  an  aliquot  part. 

RULE.  Multiply  by  the  nearest  ali<im>t  parr,  </.-• 
above,  and  add  or  subtract  the  difference  1  itm-*  the 
mi  i  iiber. 

EXAMPLE.—  131X2  times  64        864  or 

121X2  times  64  =     6400    s-  8  =  800 
1      times  64  64 


864 


EXERCISES. 


6T.     1.   Multiply  72  by  14l-2.         4.   78     hy 
2.  84  by  152/7.         £.   123  by 

54  by  17%.         n.   144  by  84  '  i 

6*8.     To  mult  inly  by   WO  an  ft   an    aliquot 
^  art  of  100. 

Annex  two  ciphers  ami  <i  <l  <1   to    the   tntin- 
f  ii     indicated    by  llic  <t  I  if/u  of  part. 


IX  AK1T1IMKT1C.  25 

EXAMPLES.—  125  times  128^  12800  -  %  of  12800 

=  16000. 
1331/3  times  36  —  3600  -  1200  =  4800. 


EXERCISES. 


0.9.  1.  Multiply  96  by  116^3.  4-  72  by  112l_>. 
120  by  137K.  o.  84  by  1142/7. 
345  by  116%.  6.  106^4  by  144. 


This  same  principle  may  be  carried  to  more  than  100 
and  an  aliquot;  to  200,  300,  and  even  to  thousands.  The 
student  will  find  much  in  this  field  for  original  investigation. 


DIVISION 


70.  To  divide  by  5. 

RULE.     Multiply  by  2  and  cut  off  one  fig  it  re. 
EXAMPLE.—  125  divided  by  5  ==  125  X  2  =  25.0. 

EXERCISES. 

71.  1.  Divide  135  by  5.         4.   265  by  5. 
&  145  by  5.         6.  325  by  5. 

3.  175  by  5.         6\   875  by  5. 

72.  To  divide  by  25. 

RULE.     Multiply  by  4  and  cut  off  two  figures. 
EXAMPLE.—  125  divided  by  25  ==  125  X  4  =  5.00. 

EXERCISES. 

73.  1.   Divide  275  by  25.         4.  875  by  25. 
£.  325  by  25.         5.   925  by  25. 

475  by  25.         6.   975  by  25. 

74.  To  divide  by  125. 

R  ULE.     Multiply  by  8  and  cut  off  three  figures. 
Ex.—  375  divided  by  125  ±=  375  X  8  ^  3.000. 


L\     ARITHMETIC.  27 

EXERCISES. 

M.     1.   Divide  500  by  125.          S.   875     by  125. 
625  by  125.          4.    1125  by  125. 

;6*.     To  dh'ide  ht/  an  aliquot  part  of  10O. 


Mn  hi  ply  by  the  denominator  of  the  frac- 
tion expressing  the  alf<{n»t  part,  Divide  In'  tJie 
numerator  and  a/r  off'neo  iigurcs. 

EAMFLES.—  240  -r-  5  =  240  X  20  —  48.00. 
840  H-  25  :=  840  X  4  —  33.60. 
1200  •*-  12^  ==  1200  X  8  =  96.00. 
1350  -f-  16^/3  ==  1350  X  6  =  81.00. 

EXERCISES. 

;;.     Divide  245  by  25.         820  by  S 

268  by  20.         725  by  83^. 
475  by  33^3      446  by  125. 


<H.     To  rtiriflr  Inj  1O<  1OO,  1OOO,  etc. 

RULE.      Cur    off    <i^    iiKiny    figures    as    there    are 
in   t  lie  <lirf*or. 


EXAMPLE.—  1240  divided  by  100        12.40, 

7.9.     To  wilnce  tlte  dirt'sor  1o  sotue   tttun- 
of  tens,  hnndwls,  thousands,  etc. 

RULE.  Multi-ply  both  divisor  and  dividend  by 
some  nmnhcr  that  ?.'/'//  make  the  divisor  a,  multiple 
of  tens,  Jin  ml  reels,  t  it  cnisa  mis,  etc.,  and  divide  as  in 
short  division. 

EXAMPLE.—  15)  2365 

2  '        2 

3.0)47370 

157  and  10  rein. 

XO'l'K.      Divulo  the  remainder  I'D  hv  '2  to  find   the  true  remainder. 

EXERCISES. 

80.     1.   Divide  3845  by  35.         5.   8732  by  75. 
*.  6492  by  45.         .$.   6288  by  125. 


28  SHOUT    MKT110IIX 

DIVISIBILITY  OF  .NUMBERS. 

81.  To  tell  when  a  number  is  fit  risible  bt/ 
2,  3,  4,  3,  6,  8,  9,  10,  etc. 

82.  All  numbers  are  divisible  by  2  when  they  end  in 
0,  2,  4,  6,  or  8. 

83.  By  3  when  the  sum  of  their  digits  is  divisible  by  3. 

84.  By  4  when  the  two  right  hand    figures  express  a 
number  divisible  by  4. 

So.     By  5  when  they  end  in  0  or  5. 
86*.     By  6  when  divisible  by  2  and  3. 

87.  By  8  when  the  three  right  hand  figures  express  a 
number  which  is  divisible  by  8. 

88.  By  9  when  the  sum  of  their  digits  is  divisible  by  9. 
8,9.     By  10  when  they  end  in  0. 

90.  By  7  or  11  if  they  consist  of  four  figures,  the  first- 
and  fourth  identical  and  the  second  and  third  ciphers. 

91.  By  any  composite  number  if  divisible  by  all  of  its 
prime  factors. 

CANCELLATION- 

92.  Cancellation  is  a  method  of  dividing    by    re- 
jecting equal  factors. 

RULE.  Cancel  any  or  «11  factors  common  to 
both  dividend  and  divisor.  Ih'cide  tlta  product  of 
those  remaining  in  the  dividend  by  (lie  product  of 
those  remaining  in  the  divisor. 

EXAMPLES.—  42  X  36  ^  24  X  14  ==  ? 
Arrange  the  numbers  as  follows  : 

03  ;5 

At  30         9 


U     X     U         2 

'1  £ 

EXERCISES. 

9:$.     1.   Divide  84     times  72       by  36     times  21. 

144  times  216     by  56     times  128. 
3.  512  times  1728  by  144  times  216. 


j  FRACTIONS 


.     To  a  d  ft  ff(t  ction*  lt<iciny  ft   cot  tun  on 
denominator. 


RULE.     Add  their  iiuim-rator*  ami    write  tin 

.^11  It  oi'fr  tJir  r<->in  iitoii  denoin  iitat»r. 


EXAMPLE.—  V7  -    2/7  +  %  =  %. 
EXERCISES. 

,ai.    /.  Add  2  9     %     T(,     . 


f>6*.     To  rff7^  f*ro  fractions 
tton 


RULK.     ^Inltiplv  tlir  ^uin  of  the  denominator  s  by 

O'imnon  ninii>:r<tror   and  ivripe  the   result  over 
of  the  denom  imttin-f. 


Ex.—  i2        i3  ==  (2  -     3)  >    1  over  2X3  = 
%        %        (3        5)   X   2  over  15  =  16/i5- 

EXERCISES. 

f>;.     1.   Add  34  ^.   -7  r>ii. 

%  -  3/7-  ^    6/7         +   6/H- 

¥5  -f  %•          &   10ia     -  10/7- 


30  SHORT    METHOD* 

98.  To  add  fractions  not  having  a  com- 
mon numerator  nor  common  denominator. 

RULE.  Multiply  each  numerator  into  all  the 
denominators  except  its  own  for  new  numerators, 
and  take  the  prod^lct  of  all  the  denominators  for  a 
common  denominator,  then  add. 

EXAMPLES.—  %  +  %  =        ^jt-"         19/i5- 

12  -r  1<>         IS 


EXERCISES. 

.9.9.     ^.  Add  %  4-  y7.         A   V2        %        %• 

%-f  6/n.       4-  %-h  3/7    r  8/n. 

NOTE.  When  .several  fractions  whose  denominators  are  not  prime  to 
each  other  are  to  be  added,  reduce  them  to  their  least  common  deuominaior 
and  add. 

TOO.     To  add  mixed  numbers. 

RULE.  Add  ivhole  numbers  and  fractions  sepa- 
rately and  tlien  unite  results. 

EXAMPLE.—  8%  -f  12%. 

8    r   12  20 

%       %-='1%5  -   His 


EXERCISES. 

101.  L  Add     91/2+141/3-     4-   283/5  +  354/5. 
^.  18%-J  252/7.     5.  431/5     72y7. 

21%1275/y.     ft   66%-  231/4+  17y5. 

102.  To  subtract  fractions  h«  rinr/  a  co  tn- 


RULE.      Take   the   difference   of    tli<> 
<tinl  write  it  over  the  common  denominator. 

EXAMPLE.  —  %  minus  %  =  %  -    %. 


IX     ARITHMETIC.  31 

EXERCISES. 

1.   Solve  :     %         _  44          .;    13/15  __  li/15. 

10/13   __  5/13.          ,J.     42/_3   __    27/_3. 

To  subtract  fractions  liarina  a  com- 
mon 


RULE.  Multiply  the  (lijft'cn'iH-f  <•/'  the  <lcntnninu-- 
tors  In'  the  common  numerator  <iiid  -write  the  re- 
*u  It  over  the  pr<»l  it  ct  of  the  >1  eu<n»  7  7  ir/r»  »;-.-•. 


2    /  •> 

EXAMPLE.—  %  -  2/7  = 


EXERCISES. 

/OJ.      /.   Solve:     %  —  3/7.          ,;.    s/^      _  8/15. 

%  -  %•  5. 

~  5ll.  6'. 


*.     7V>  subtract  fractions  had  it  f/  neither 
numerators  nor  comnton  denom- 
inators. 

RULE.     Mu  I  tiply  ench  numerator  into  the   other 

(If/loin  iii'i  r<>rs,  take  the  difference  un<l  i^rite  it  over 
t  In-  pnnl  a  rr  of  the  denominators. 

I.',        l:-j 
EXAMPLE.—  %  —  3/7  -  _ 


EXERCISES. 

107.      1.    Solve  :       6/?  _  5/8.  .;. 


.     To  subtract  tni.red  numbers. 


RULE,     ^rihrract   u'h.ole    numbers   and  fractions 
^l  v,  uniriu.g 


32  SHORT     UKTUOUX 

NOTE.  If  the  traction  of  the  .subtrahend  is  i>-n-aU-r  than  tliaT  of  the  min- 
uend subtract  a  unit  from  the  minuend  and  add  it  to  tin-  traetion  before 
taking  the  difference. 

EXAMPLE.—  8%  —  5% 

8  —  5  :=  3 

%   ~    %  Vlo 


11   -  8  -      3 
IVa  ~  V2  % 

3%. 

EXERCISES. 

Solve:      22%  —  16%.     5.   89%  -  35%. 
75%  _  48%.     ^   95i/6  __  7434. 


XOTE.      A  g\.od  method  is  to  take  the  complement  of  the  diiference  of  th« 
fractions  when  the  subtrahend  fraction  is  the  greater. 

EXAMPLE. —  S1/^  —  2% 

4—2  2 

%  —  ¥2  —  VG  write  the  complement     % 


EXERCISES. 

110.     1.  Solve:  8i/i      -  51/3.         S.   25%  --  17%. 

.   2.  15%  -  4%.         4.   44%  -  313/4. 


.     To  fl-nt?  the  square  of  ff  nti.rert  num 
ber  whose  fr «et ion  is  1/2. 

RULE.     Multiply  the.  integer  by  tin-    next    li-igliei 
number  a-nd  a>miex  %. 

EXAMPLES.—  2%   X  2^  ==  2  X  3  -f-  ^  == 
3^   X  3^     -  3  X  4    <  = 


" 


IX  ARITHMETIC.  33 

EXERCISES. 

•>.      /.    Multiply  4V2  by  4V2.          &   8V2     by  8V2. 
51/2  by  5%.         ^.   9V2     by  9%. 


To  /iwd    f/if  product  of  two  wired 
n  anthers  n'ltose  fractions  arc  172. 


Rl~LE.    *To  rlif  pr<xl  n  ct  ••/'/  lir  iiii-i'gt  T.--  */«/«/  15  tln-ir 
.<ti  in  (Did  a  n.ni'.\-  L4. 


Ex.—  2i  o     C  4V2  =  2  >    4-        3  X  14== 

3V2  >:  4V2  =  3X4  -    3V2  +  ^4  =  15%. 

XOTE.     The  fraction  will  be  ooe-fparth  if  the   sain  of  the  two   integers  i 
.-ii  :  if  the  sum  is  odd  thi-  fraction  is  three-fourths. 

EXERCISES. 

114.      1.    Multiply  21/2  by  6V2.          •',.   Sifeby 
3V2  by  5V2.         4.  41/2  by 


To  find  the  ptoduct  of  two   mixed 
s  whose  integers  are   identical  and 
the  sutn  of  whose  fractions  is  a  unit. 

RULE,.      ^Multiply  the  inrrgrr  by  the.   nr.\-f    In'glirr 
TininbtT  ami  <iiuit'.\-  rlir  prodnrr  oftlie  fr<tct; 

Ex.—  2i3        2%  =23        I's  X  %  —  6%. 

3#  X  33^  =  3  >    4        ^     (  3^==  123/16. 

EXERCISES. 

/  /6\      /.   Multiply  4%  by  43/5.     ^.   9y7      by  93/7. 
?.  5%  by  5ix5.     5.   12%    by  10%. 

63/8  by  6%.     6.   153/n  by  158/ii. 

I  IT.  To  find  the  product  of  two  n  anthers 
whose  integers  are  consecutive  and  the  snnt 
of  whose  fractions  is  a  unit. 


34  SHORT    METHODS 

RULE.  Multiply  the  greater  number  increased 
by  1,  by  the  less;  and  for  the  fraction  annex  ilir 
complement  of  the  square  of  the  fraction  of  tli-r 
greater  number. 

EXAMPLE.—  4/3  X  3^  =  5  X  3  -f-  %  ?=  15%. 

XOTK.  The  .square  of  one-third  equal.-*  one-ninth,  its  complement  is 
eight-ninths. 

EXERCISES. 

118.  1.   Multiply  514  by  4%.     4.   9%      by  8%. 
2.  63/5  by  5%.     5.   12%    by  11%. 
8.                   83/7  by  7y7.     «.   205/12  by  19%2. 

119.  To  find  the  prod  net  of  tico   iHi.red 
numbers  whose  inteyei's  are  identical. 

RULE.  To  the  product  of  the  Integers  add  tlic 
product  of  the  siim  of  the  fractions  times  tJic  com- 
mon integer  and  the  product  of  the  fraction*. 

Ex.—  6y2   X  6l/3  =  6  X  6  +  6  X  %  +  %  X  #  - 
36  +  5  -:    V6  =~  41^. 

EXERCISES. 

120.  1.   Multiply  81X2     by  S^i         4.   24%  by  24%- 
«.  121/3  by  12%.       o.   351/5  by  353/5. 
8.                   142/7  by  146/7.       6.   45%  by  45%. 

121.  To  find  the  product   of  ttro   mixed 
numbers  when  the  fractions  are  identical. 

RULE.  To  tlic  product  of  the  integers  add  the 
product  of  the  sum  of  the  integers  times  the  com- 
mon fraction  and  the  product  of  the  fractions. 

Ex.—  41/3  X  8i3  =  4  X  8  X  12  X  1X3  -   1X3   X   173 
32       4        i' 


EXERCISES. 

122.     1.   Multiply  6^  by  18^.         .1    36178  by  441/8- 
3.  91/3  by  15%.         4.    721/9  by  36%. 


WO. 


IX     .\RIT1I  MET  1C.  35 

To  multiply  by  (in    aliquot  pftrt   of 


RULE.     AII.II.CJ;  two  ciphers   to   the   multiplicand 

nnd  tube  such  a   part  of  it    as   the   multiplier   is    a 
part  of  100. 

EXAMPLE.—  24   <  16%  —  2400  ~6  —  400. 

EXERCISES. 

124.     1.   Multiply  39  by  33V3.         4-   54    by  66%. 

2.  48  by  12>2  5.  72    by  37  1,> 

64  by     8V3.         fj.   144  by  83V3. 

12Z.     To  Multiply  a  f  fact  to  it    by   a  frac- 
tion. 

RULE.  Cancel  all  common  factors  in  numer- 
ators and  denominators  and  divide  the  product  of 
those  remaining  in  the  numerator  by  the  prod^lct 
of  those  in  the  denominator. 

EXAMPLE.  —   3          0         4          :\          :\ 

_      \/      ___      \/       ___       N,/       ^__     --      _ 

4       26  7       2i0        2S 

EXERCISES. 

126.     1.     Multiply  %  by  %      by  %5. 
2.  %  by  2i/25  by  27/32. 

7^7.     To  fit  ride  a  fraction,  by  a  fraction. 

RULE.  Invert  the  divisor  and  proceed  as  in 
multiplication  of  fraction  *. 

EXAMPLE.—  %  X  %  +  7/io  X  9/i6  = 

2 

3  4        10        1C)       :-l-2    11 

—  X  —  X—  X—  ~     —  1  — 

4  $          7        -03       -21     L>1. 

EXERCISES. 

128.     L     Solve  :  %  X  7/10  X  %  -f-  2V24  X  15/28. 

^.  6/7   X    H12  -*-  22/49    X    %   X    %• 


PERCENTAGE 


119.     To  find  the  percentage  when  the  rate 
is  an  aliquot  part  of  200. 

RULE.     Take  such  a   part  of  the  iimnbcr  as    tlie 
rate  is  a  part  of  100. 

EXAMPLE.  —  12M  per  cent  of  64  ==  }i  of  64  =  8. 

EXERCISES. 

13O.     1.  Find  50  per  cent,  of  38.  Of  346. 

2.  33^     "        "  42.  Of  543. 

8.  160     "        "  96.  Of  186. 

4-  12#     "       "  128.  Of  4168. 


131.     To  find   the    percentage    tc/teit    the 
rate  is  an  aliquot  part  of  WOO. 


the  number  by  10,  and  ta-hr.  *ucli  a  part 
of  it  as  the  rate  is  a  part  of  1000. 

Ex.—  830  per  cent  of  144  =  17i2  of  1440  ^  120. 

EXERCISES. 

132.     1.  Find  333  ^  per  cent  of  27.  Of  279. 

&  1660       "         "  66.  Of  576. 

3.  83V3                    "  96.  Of  3612. 

4-  62^          "         "  288.  Of  1624. 


fX    ARITHMETIC.  37 

133.  To    find    the    percentage    when    the 
rate  is  ant/  number. 

RULE.     ^Multiply  the  base  bv  the  rate  and    point 

off  tu'o  places. 

Ex.—  12  per  cent  of  $400  =  400    <  .12  =  $48.00. 

EXERCISES. 

134.  1.      Find  15  per  cent  of  500-          Of  1879. 

22       "        "   750.         Of  4321. 

18       "        "   560.         Of  8765. 

4.  27       "        "   1340.       Of  9876. 

135.  To  find  the  bff.se ,  the  rate   an<l  per- 
centftf/e  briny  f/iren. 

KTLE.      Divide  the  pereenrage  !,y  the  rate. 

EXAMPLE. —  Rate  =  12  per  cent,  Percentage  ~  96. 
96  -f-  .12  —  800  Base. 

EXERCISES. 

13<>.      1.  Rate  4     per  cent,  Percentage  52,     Ba.^e         ? 

2       "    9        "  "          144      "     ? 

.;.      "    12      "  176      "          ? 

13  7.     To  find  the  rate,  the  percentage  and 
base  behif/  f/iren. 

KULE.      Divide  tlte  percentage  by  rlie  base. 

EXAMPLE. —  Base  =  400,  Percentage  =  36. 
36  -T-  400  =  .09,  or  9  per  cent. 


EXERCISES. 

138.     1.  Base  500,     Percentage  35,     Rate  =     ? 
«.      "    1200,  "         72,        "    =  ? 

-;.      "    1800,  144,      " 


38  SHORT   METHODS 

139.  To  find  the  rate  of  loss  or  t/nhi. 

RULE.     Divide  the  loss  or  gain  by  the  cost. 

EXAMPLE.—  Cost  ==  $250,  Selling  price  =  $300. 
$300  —  $250  =  $50,  Gain, 
$50  -^  $250  =*  20  per  cent.,  rate  of  gain. 

EXERCISES. 

140.  Find  Rate  of  Gain  or  Loss  : 

1.  Cost  =  $400,  Selling  Price,  $500. 

2.  "     ^$279,  "          $540. 
2.      "       =  $720,  "          $600. 

14,1.     The  following  formulas  are  a  very  good    illustra- 
tion of  the  problems  of  percentage  : 

FORMULAS  OF  PERCENTAGE. 

Base  X  Rate  ==  Percentage. 
Percentage  -~  Base  =  Rate. 
Percentage  -~  Rate  ==  Base. 

Amount  -f-  1         Rate  :=  Base. 
Difference  -=-  1  --  Rate  ±    Base. 

By  applying  the  formulas    above    to    these    applications, 
problems  of  Percentage  are  very  readily  solved. 


IX  ARITHMETIC. 


39 


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INTEREST 


r .  ••"• 


CANCELLATION  METHOD. 

'143.     EXAMPLE. —  Find  the   interest  on    $420  for  30 

days,  at  7  per  cent. 


35 


$00     30  days 

.07 
$2.45  interest.      A  us. 

EXAMPLE. —  Find  the  interest  on  $540  for  7  months  at 
9  per  cent. 


45 

7  months 
.09 


$28.35  interest.      A-ns. 

RULE.  \Vrite  the  principal,  rate  and  time  </&•  »/• 
dividend,  and  one  year  expressed  in  the  same  de- 
nomination as  the  time  gii'eii.  as  <i  divisor,  cancel 
and  reduce. 

EXERCISES. 

144.  Find  the  interest : 

/.  Of  $1200  for  42  days  at  6  per  cent. 

2.  Of  $1800  for  33  days  at  5  per  cent. 

3.  Of  $2250  for  60  days  at  1  per  cent. 

4.  Of  $8400  for  5    mos.  at  8  per  cent. 

5.  Of  $9600  for  9    mos.  at  6  per  cent. 

6.  Of  $9636  for  1  year,  4  mos.  at  7  per  cent. 


7.V  ARITHMETIC. 


To  find  tin'  interest  tr/ten  the  tune  is 
in  months  ft  no1  flays. 


EXAMPLE.  —  \Vhat  is  the  interest  on  $240  for  3  months, 
12  days  at  6  per  cent. 

3  months,  12  days  102  days,  or, 

3  months,  12  days  3.4  months. 

3    $240  '-'  2    $240  120 

10-2  days  12     :j.4  months 

.06   2  .0* 


$4. OS  interest.          $4. OS  interest. 

RULE.  7V«o'/ </  a-s  //*  cancellation  method,  rc- 
•  hi rin ^  r//<  time  r<>  </<M'>,  or  r<»  nnmrli.--  <nul  tenths  <>f 
(t  month. 

XOTE.  When  the  number  of  day*  is  ;t  multiple  of  I*  it  shorten-  th,- 
vvork  hv  using1  months  ;\nd  tenths  of  ;i  month. 


ABBREVIATED  METHOD. 


The  cancellation  method  may  be  somewhat 
shortened  by  omitting  the  rate  and  using  instead  of  360  as 
a  divisor  the  quotient  of  the  rate  into  360.  Thus  : 


When  the  rate  is 

2 

per  cent. 

use 

180. 

;  (                   tt 

3 

•• 

(i 

120. 

•  . 

4 

ti 

90. 

a 

5 

tt 

72. 

.. 

6 

n 

•• 

60. 

a                   ti 

8 

(I 

a 

45. 

tt                           it 

9 

tt 

it 

40. 

a 

10 

tt 

a 

36. 

a 

12 

•• 

30. 

18       "  "    20. 


4a  *1IORT    METHODS 

EXAMPLE. —  What  is  the  interest  on    $720  for  33  days 
at  5  per  cent? 


10 


$:>.::>()  interest. 

EXAMPLE. —  What  is  the  interest  on  $1260  for  66  days 
at  8  per  cent  ? 

1$     \       85 
$1200 
0022 


$18.70  interest. 

EXERCISES. 
147.      Find  the  interest  : 

1.  Of  $840     for  18  days  at  6  per  cent. 

2.  Of  $960     for  27  days  at  8  per  cent. 

3.  Of  $1240  for  36  days  at  4  per  cent. 

4.  Of  $3260  for  63  days  at  9  per  cent. 


BANKERS'  METHOD. 

148.     EXAMPLE. —  What  is  the  interest  on   $1344  for 
75  days  at  6  per  cent  ? 

$13.44  =  interest  for  60  days. 

3.36        interest  for  15  days. 

$16  80  =  interest  for  ~75~days. 

IiULE.  Point  off  two  places,  ivhich  -will  give  the 
interest  for  the  rate  and  corresponding  time  as 
fol ioius  : 


IX    ARITHMETIC. 


'2     per  ceu  i  for  ltf(. 

3 

4 

90 

5 

72 

6 

60 

- 

45 

.</ 

40 

10 

36 

12 

30 

IS 

20 

Then  take  such  nHquot   part* 
v  iiee<led  f<»'  the  gii<en  time. 

«/</vs. 


>f   t/// 


EXERCISES. 

Find  the  interest  : 

/.      Of  $810     for  90  days  at  4     per  cent. 

Of  $648     for  45  days  at  8     per  cent. 

Of  $1232  for  36  days  at  10  per  cent. 

Of  $7200  for  37  days  at  9     per  cent. 

Of  $963.75  for  80  days  at  6  per  cent. 
o.  Of  $2140.50  for  90  days  at  8  per  cent. 
V.  Of  $5235.60  for  66  days  at  6  per  cent. 
7.  Of  $4840.40  for  72  days  at  10  per  cent. 


PROBLEMS  I.N  INTEREST. 

1.5O*  The  following  formulas  are  illustrative  of  the 
four  problems  of  interest. 

4-  Principal    >    Rate    •    Time        Interest. 

.',.  Interest  -~   Principal         Rate  ~=  Time. 

.'.  Interest    :    Principal         Time  =  Rate. 

/.  Interest    ~  Time    ~    Rate        Principal. 

131.  Applications  of  Percentage  involving  the  ele- 
ment of  time  are  as  follows :  Interest,  Discount.  Partial 
Payments,  Insurance,  and  Stock  Investments. 


44 


XII ORT     METHODS 


To  find  the  timv  when  t/te  principal, 
rate  and  interest  is  given. 

EXAMPLE. —  Principal   -     $900;    Rate  =  8    per    cent.; 
Interest,  $6.00;  to  find  the  Time. 


300 

$<>.00 


$000 


oO  days,  the  tune. 


EXAMPLE.  —  Principal  $720;  Rate  6  per    cent.;  Interest 
$25.20.     Find  the  time. 

n       00 


(?) 
.00 

7  months. 

RULE*  Use  the  cancellation  inetJiod  </<s  in  reck- 
oning interest,  using  the  product  of  the  interest 
and  one  year  expressed  in  the  proper  denomination 
as  a  dividend  and  the  product  of  the  principal  and 
rate  as  a  divisor. 

133.  To  find  the  rate  when  the  principal 
time  and  interest  are  f/iveti. 

EXAMPLE.  —  Principal,  $960;  Time,  45  days;  Interest, 
$8.40.  Find  the  rate. 

3  ' 

360  $000  ^ 
7    10$          ^  days  1$ 
Interest  M0  :   (?) 

An*.    7  per  cent.,  Rate 

EXAMPLE.  —  Principal,  $1050;  Time,  3  months:  Inter- 
est, $21.00.  Find  the  rate. 


4    12 


21.00 


3  months 


Ans.   8  per  cent.,  Rate. 


IX  ARITHMETIC.  45 

KULK.     Same  as  for  152,  except  that    the  product 
of  thr  Principal  and  Time  is  used  as  a  divisor. 


.     To  find  the  pfincijtft/.  the  rate,  time 
and  interest  beiny  f/iren. 


300  [  ?  ] 

-5  60  days 

Interest  $5.23  .07 

An*.  *450,  Principal. 


3 

12 
Interest  S45. 00 

2.50  f,  00 


9  month: 


.   *750,  Principal. 


RULE.     Same  as  for  15'J,  «'.\->:i'pt  that    tJi>' 
Tim?  and  Rut?  is  us?<i  as  a  difisor. 


L>,>.     To  find  the  Rank  Discount  of  any 

sum. 

EXAMPLE.^; Find  the  bank  discount  of  $840  for  63  days 
discounted  at  bank  at  10  per  cent. 

$?<40   70 

03  21 
300      10' 
I 


An*.   $14.70  bank  discount. 

ItL'LK.     Kititl  tin1   sfmpl?    iii.r.?r?st.    f»r    the   given 

tiii)<'  an  1 1   r<t  t  r. 


To  fin  tl  the  True    Discount   of  <tnt/ 


EXAMPLK.  -What  is  the  True  Discount  and  present 
worth  of  a  debt  of  §530,  due  in  one  year,  discounted  at  6 
per  cent? 

#530    :    1.06        $500  the    present  worth; 
$530  —  $500        $30  the  true  discount. 


/vf'LK.  Divide  the  aiivount  of  the  debt  by  I  />//<> 
the  rate  for  the  given  time  t  th<i*  i^i-ll  give  'tin-,  pres- 
ent worth;  subtract  the  present  luorth  from-  th<j 
debt,  tlie  difference  is  f/ir  t  rue  discoutit. 


ANALYSIS. 


The  /i wtt  step  in    analysis  is  to   rp<hir<>    to    tin' 
unit  as  follows: 

If  4  hats  cost  $20,  1  hat  will  cost  ^  of  $20,  or  $5. 

The  second  step  is  to  reduce  to  a  number: 
If  1  hat  cost  $5,  7  hats  will  cost  $35. 

The  third  step  combines  the  first  and  second: 
If  7  coats  cost  $84,  1  coat  will  cost  $12 ;  4  coats   will 
cost  $48. 


EXERCISES. 

258.     If  13  hats  cost  $39,  what  will  7  hats  cost? 

2.  If  11  pairs  of  shoes  cost  $46.50,  what  will    7   pairs 
cost? 

3.  If  y*  of  a  ton  of  hay  cost  $10,  what  will    J^   of  a 
ton  cost? 


47 

15  &•  Reduce  the  following  first  to  the  fractional  unit, 
then  to  the  integral  unit,  then  to  the  required  number  of 
fractions. 

EXAMPLE. — If  %  of  a  ton  of  hay  cost  $12,  what  will  fe 
of  a  ton  cost  ? 

%  of  a  ton  cost  $12, 

l/5  of  a  ton  will  cost  $3, 

%  or  1  ton  will  cost  $15, 
r/6  will  cost  /s  of  15  or  *%, 
~>0  will  cost  7'times  i5/8=i05/8=: 


EXERCISES. 

1(>0.      /.      If  73  of  a  bushel  of  wheat  is  worth  72  cents, 
what  are  10  bushels  worth? 

2.     If  9/io  of  an  acre  of  land  cost  $108,  what  will  %  Of 
an  acr*e  cost  at  the  same  rate? 

.}.   If  y?>  of  ^  of  a  cord  of  wood  is  worth  $3.50,  what  is 
of  %  of  a  cord  worth  ? 


1(>1.     To  find  interest  on  overdrafts. 

EXAMPLE. — Overdrafts  for  the  week  were  as  follows  : 

7.      1200 
1500 


Interest  at  10  per  cent. 
IbOU 

1600 
1850 


9500  -*-  360/10       $2.64. 


RULE.  Divide  the  sum  of  the  daily  overdrafts  by 
360  divided  by  tlie  rate,  and  point  off  two  decimal 
places. 


48 


102.     How  to  find  ert-ors  s/towtt  by  a  trial 
balance. 


I.  See  that  your  former  baln-ncc  of  Imlnm'cs  is  in 
balance. 

;„'.      Be  sure  that  your  additions  are  correct. 

Find  the  exact  amount  out  of  balance,  and  look  for 
it  and  its  one-half  among  the  ledger  items. 

Jf.  If  the  error  is  9  or  a  multiple  of  9,  look  for  reversed 
figures. 

EXAMPLE.  —  65  written  56  would  make  a  difference  of 
9  ;  57  written  75  would  make  a  difference  of  2  times  9,  or 
18  ;  63  written  36  would  make  a  difference  of  27,  etc. 
This  may  occur  in  any  or  all  columns. 

5.  If  there  is  an  error  of  1  in  any  column,  look  tor  er- 
rors in  addition. 

#.  If  the  error  is  small,  look  for  it  in  Interest  or  Dis- 
count. 

7.  Examine  the  Bills  Receivable  and  Bills   Payable  ac- 
counts and  note  that  the  Debit  and    Credit   entries   are   ex- 
actly alike  as  far  as  posted. 

8.  See   if  your   cash    account   in   the   Ledger   or  Cash 
Book  agrees  with  your  Banking  Ledger  and  cash  on  hand. 

9.  If  the  error  is  in  cents  column,  it  is  not  necessary  to 
add  the  dollars  column. 

10.  If  the  above  tests  will  not  indicate  to   you   the  er- 
rors, it  will  be  necessary  for  you  to  re-check  everything  from 
the  previous  balance  of  balances.      Do  not  go  over  the  work 
without  checking,  you  will  .waste  your  time  if  you  do. 


J/ 


U.  C.  BERKELEY  LIBRARIES 


